MIMO Communications and Algorithmic Number Theory

1. MIMO Communications and Algorithmic Number Theory G. Matz joint work with D. Seethaler Institute of Communications and Radio-Frequency Engineering Vienna University of Technology (VUT)

2. Setting the Stage • MIMO communications: • Algorithmic number theory (ANT) is the study of algorithms that perform number theoretic computations(Source: Wikipedia) Examples: primality test, integer factorization, lattice reduction channel . . . . . . TX RX TX antennas RX antennas

3. Outline • MIMO detection and ANT • MIMO precoding and ANT • Precoding via Vector Perturbation • Approximate Vector Perturbation using Lattice Reduction • Lattice Reduction using ANT: Brun‘s Algorithm • Simulation Results • Conclusions

4. Multi-Antenna Broadcast (Downlink) user #1 System model: user #k channel . . . precoding . . . user #K M TX antennas K symbols users, each with one antenna Users cannot cooperate shift MIMO processing to TX precoding MIMO I/O relation: with CSI at TX required

5. Vector Perturbation (Peel et al.) TX vector: • here, and is an integer perturbation vector • precoder performs channel inversion and vector perturbation Receive symbols: • follows from • RX-SNR equals 1/choose z such that s(z) is “short“ • Remaining RX processing: • get rid of z via modulo operation • quantization w.r.t. symbol alphabet

6. Choice of Perturbation Vector • Optimum vector perturbation maximizes RX-SNR: • Implementation: sphere encoder • Suboptimum precoding: e.g. Tomlinson-Harashima precoding (THP) • For channels with large condition number • sphere encoding has high complexity • THP etc. have poor performance • Small condition number: all methods work fast and well

7. 4 4 1 1 0 0 3 3 1 0 0 2 2 10 1 1 1 1 0 0 6 6 0 0 8 8 0 0 0 2 0 4 0 1 0 0 0 2 0 4 0 1 0 0 c o n d i t i o n n u m b e r Structure of Channel Singular Values M=K=4 smallest singular value and associated singular vector v cause problems

8. Vector Perturbation for Poor Channel Condition Example: BPSK, real-valued channel & noise, M=K=2, =2 search TX vector that - is integer - has small length - is orthogonal to v approximate integer relation (ANT) TX vector perturbed versions of TX vector

9. Relation of MIMO and ANT MIMO Detection at Rx Precoding at Tx duality poorly conditioned channels ANT Approximate Integer Relations Simultaneous Diophantine Approximations duality

10. Vector Perturbation Using Lattice Reduction (LR) • View as basis of a lattice • Try to find “better“=reduced lattice basis • All lattice basis are related via a unimodular matrix: • LR-assisted vector perturbation ( , ) • cost function: • solve or use THP approximation • use as perturbation vector

11. Lattice Reduction • Orthogonality defect (quality of lattice basis): • LR: find achieving small and thus small left channel singular vectors channel singular values • Most popular LR method: Lenstra-Lenstra-Lovász (LLL) algorithm • LLL-LR assisted THP achieves full diversity • but LLL can be computationally intensive

12. Integer Relation Based LR • Goal: more efficient LR method • To achieve small , vectors must be • sufficiently orthogonal to singular vectors with small singular values • For poorly conditioned channels, only one singular value is small find integer vectors that are sufficiently orthogonal to • This is the approximate integer relation (IR) problem in ANT • IR-LR focuses on one singular vector (in contrast to LLL-LR) - some performance loss - significantly smaller complexity

13. Approximate Integer Relations • can be made arbitrarily small using long vectors • large will increase • Tradeoff: • small governed by channel singular values • small • Approximate IR: achieve small with as short as possible • Can be realized very efficiently using Brun‘s algorithm

14. Brun’s Algorithm • Initialization: • Find • Calculate repeat until termination condition is satisfied • Replace (update of ) • Very simple:scalar divisions, quantizations, and vector updates • is also updated recursively and can be made arbitrarily small

15. Performance of Brun’s Algorithm • Example using and averaging over 1000 randomly picked average average no. of iterations

16. Lattice Reduction via Brun’s Algorithm Termination condition • at each iteration, is a basis for • recall: LR aims at minimizing terminate if update of does not decrease Calculation of • we are just interested in channels with one small singular value • in this case, apply Brun’s algorithm to any column of

17. Simulation Results (1) • iid Gaussian channel • 4-QAM • Iterations on average: • Brun: 2.5 • LLL: 12.9 THP w. LLL Symbol Error Rate THP • A Brun iteration is less • complex than an • LLL iteration THP w. Brun Sphere encoding (optimal) SNR LR using Brun‘s algorithm can exploit large part of available diversity

18. Simulation Results (2) • iid Gaussian channel • 4-QAM • Iterations on average: • Brun: 4.8 • LLL: 42 Symbol Error Rate THP w. LLL THP • A Brun iteration is less • complex than an • LLL iteration THP w. Brun Sphere encoding (optimal) SNR LR using Brun‘s algorithm can exploit large part of available diversity

19. Conclusions • Algorithmic number theory provides useful tools for MIMO detection and MIMO precoding • Here: proposed vector perturbation using lattice reduction based on integer relations • Efficient implementation: Brun‘s algorithm • Good performance at very small complexity

20. MIMO Detection • RX vector: • ML detector: • exact implementation: sphere decoder • suboptimum detectors: ZF, MMSE, V-BLAST, … • If ispoorly conditioned: • poor performance (ZF, MMSE, V-BLAST, ...) • or high complexity (ML) • Everything is fine if is close to orthogonal

21. Detection for Poor Channel Condition Example: BPSK, real-valued channel & noise, M=K=2 v search TX vector that is - integer - close to line y+v simultaneous Diophantine approximation (ANT) ZF-domain Rx vector: